Leibniz (Fraction) Notation Exercises

Example 1

Example 2

For the integral, (a) identify u and du and (b) integrate by substitution.

Answer

(a) We have

u = 6x

du = 6dx

(b)

Example 3

For the integral, (a) identify u and du and (b) integrate by substitution.

Answer

(a)

u = x + 1

du = dx

(b)

Example 4

For the integral, (a) identify u and du and (b) integrate by substitution.

Answer

(a)

u = (x2 + 3x)

du = (2x + 3)dx

(b)

Example 5

For the integral, (a) identify u and du and (b) integrate by substitution.

Answer

(a)

u = e6x

du = 6e6xdx

(b)

Example 6

Integrate by substitution.

Answer

We have

u = (5x + 7)

du = 5dx

If we break up 60 into 12 ⋅ 5, we can see how to do the substitution.

Example 7

Integrate by substitution.

Answer

We have

u = (7x2 + 11)

du = 14xdx

So

Example 8

Integrate by substitution.

Answer

We have

u = (2x2 + 4x)

du = (4x + 4)dx

It doesn't look like there's a factor of (4x + 4) in the integrand. However, (4x + 4) is the same thing as 4(x + 1). If we factor 60 into 15 ⋅ 4, we can see where the substitution happens:

Example 9

Integrate by substitution.

Answer

We choose u to be the quantity in the denominator, since we know how to integrate but we don't know how to integrate . So

u = sin (e4x)

du = 4excos(e4x)dx

Then we factor 16 into 4 ⋅ 4 and do the substitution:

Example 10

Integrate by substitution.

Answer

We have a couple of choices here. One possibility is

Then

Another possible choice is

Then

Thankfully, we get the same answer regardless of which substitution we use. We can also have integrals where a constant factor of du is missing. In such cases we can multiply by a clever form of 1 to introduce the missing factor to the integrand.

Example 11

Integrate.

Answer

Take

u = 4x2 + 6x

du = (8x + 6)dx

= 2(4x + 3)dx

The integrand is missing a factor of 2, but we can fix that.

Example 12

Integrate.

Answer

Take

The integrand is missing a factor of , so we multiply the integrand by and the integral by 2:

Example 13

Integrate.

Hint

Simplify the integrand.

Answer

This one doesn't require substitution. Following the hint, we simplify the integrand to get

We could use substitution instead, but it would take us more work to get the same answer. Take

u = 4x

du = 4dx

Then we have to multiply by a clever form of 1:

Example 14

Integrate.

Answer

Take

u = 16x2 – 10

du = 32xdx

We almost have the derivative of u in the numerator of the fraction, but we're missing the factor 32. So we multiply the integrand by 32 and the integral by .